Suppose now that we can identify one or more population groups whose membership is fixed over time. The cross sectional variances in equation (8) can then be estimated by computing the corresponding sample variances within groups at different time periods. This application of synthetic panel techniques yields consistent estimates of the corresponding population variances. We can then run the regression:
and test the hypothesis that /г= 1.
The use of OLS to estimate equation (9) would produce inconsistent estimates of 7Г, however, for at least three reasons. First, the hypothesis that the cross-sectional covariance of Avi /+l with the observed component of marginal utility at time t is zero is likely to be violated. Second, the time varying-terms included in GJl+j might be correlated over time with the variance term in equation (8). Finally, in a finite sample one never observes the true population moments, but only an estimate of these moments which are therefore affected by measurement error. This induces a further source of bias in the estimate of к.

The issue is therefore that of finding an instrument for var[ln(c(,) + # z,,]which is uncorrelated with GJt+I and with the sampling error. Synthetic cohort data suggest an ideal candidate for dealing with the potential bias arising from measurement error: if the samples used in estimating the variances in equation (9) are independent – as they typically are in repeated cross sections – so are the sample errors of subsequent periods.

Therefore, var[ln(c(;_j ) + #z.f4] is, as far as sampling error is concerned, a valid instrument for the corresponding variance at time t with which it is obviously correlated. To guarantee that the (twice) lagged variance is a valid instrument overall, however, one must assume that covj^^varfln^,,!)-}-# z, ,_i]|=0, i.e. that the twice lagged variance is uncorrelated with the deviation of cot+i from its unconditional mean. This is a strong assumption that deserves further scrutiny.

To better understand the issues involved, it is worth to strip the model down to its simplest version, the one with certainty equivalence behind equations (1) and (2), where 0)t+] – var(£if+1) + cov(cf,,£./+1). If aggregate shocks are part of agents’ information set, the second term is a genuine innovation and therefore its covariance (over time) with the lagged cross sectional variance is zero under the permanent income hypothesis. The assumption that var(c,-.,) is uncorrelated with the deviations of var($.,+/) from its time mean, however, is not an implication of the permanent income hypothesis. Rather, its validity depends on the nature of the shocks £iJ+} , on agents’ information set as well as on the market structure in which they operate.